Step |
Hyp |
Ref |
Expression |
1 |
|
nvs.1 |
|- X = ( BaseSet ` U ) |
2 |
|
nvs.4 |
|- S = ( .sOLD ` U ) |
3 |
|
nvs.6 |
|- N = ( normCV ` U ) |
4 |
|
recn |
|- ( A e. RR -> A e. CC ) |
5 |
4
|
adantr |
|- ( ( A e. RR /\ 0 <_ A ) -> A e. CC ) |
6 |
1 2 3
|
nvs |
|- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) ) |
7 |
5 6
|
syl3an2 |
|- ( ( U e. NrmCVec /\ ( A e. RR /\ 0 <_ A ) /\ B e. X ) -> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) ) |
8 |
|
absid |
|- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A ) |
9 |
8
|
3ad2ant2 |
|- ( ( U e. NrmCVec /\ ( A e. RR /\ 0 <_ A ) /\ B e. X ) -> ( abs ` A ) = A ) |
10 |
9
|
oveq1d |
|- ( ( U e. NrmCVec /\ ( A e. RR /\ 0 <_ A ) /\ B e. X ) -> ( ( abs ` A ) x. ( N ` B ) ) = ( A x. ( N ` B ) ) ) |
11 |
7 10
|
eqtrd |
|- ( ( U e. NrmCVec /\ ( A e. RR /\ 0 <_ A ) /\ B e. X ) -> ( N ` ( A S B ) ) = ( A x. ( N ` B ) ) ) |